Fast Computation of Polyharmonic B-Spline Autocorrelation Filters

Abstract

Abstract-A fast computational method is given for the Fourier transform of the polyharmonic B-spline autocorrelation sequence in dimensions. The approximation error is exponentially decaying with the number of terms taken into account. The algorithm improves speed upon a simple truncated-sum approach. Moreover, it is virtually independent of the spline's order. The autocorrelation filter directly serves for various tasks related to polyharmonic splines, such as interpolation, orthonormalization, and wavelet basis design.

Fractional Laplacian pyramids

“…The delicate step is the precalculation of Hγ(e jω ) and H2γ(e jω ), which can be done efficiently thanks to a recently published result that exploits the connection between these frequency responses and the Epstein zeta function [8]. Fig.…”

“…We first present some experimental results comparing the Burt and Adelson Laplacian pyramid (LP) and the fractional Laplacian pyramid (fLP) described in (8) in terms of energy compaction over 2D images. Pyramid decompositions of Lenna (512 × 512) are shown in Fig.…”